
It is easy to see that \(\theta \)  is a Maurer-Cartan element in the graded Lie algebra \(\big ( \oplus _{n \ge 0} \mathrm {Hom}(\wedge ^{n+1} \mathfrak {l}, \mathfrak {l}), [~,~]_\mathsf {NR} \big )\) . Therefore, it induces a differential \(d_\theta := [\theta , -]_\mathsf {NR}\)  on the graded vector space \(\oplus _{n \ge 0} \mathrm {Hom}(\wedge ^{n+1} \mathfrak {l}, \mathfrak {l})\) . Hence by the derived bracket construction of Voronov [1]}, the shifted graded space
\(\mathfrak {a}[-1] = \oplus _{n \ge 1} \big ( \mathrm {Hom}(\wedge ^{n} \mathfrak {g}^{\prime }, \mathfrak {g}) \oplus \mathrm {Hom}(\wedge ^{n-1} \mathfrak {g}^{\prime } \otimes \mathcal {V}^{\prime }, \mathcal {V} ) \big )\) 
